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Lecture 1 | Introduction to Scattering Amplitudes

NIMA ARKANI-HAMED: I'll be giving a series of lectures on the physics of scattering amplitudes, largely in [INAUDIBLE] theories, and largely in n equals 4 [INAUDIBLE]. And I just want to start off giving a brief motivation for why these questions are interesting. This is a subject that I've come to feel very, very strongly about its importance, over the past couple of years. And it's also a subject that's been developing, just at an absolutely explosive pace over the past five, seven years, and the past two or three years particularly. And I want to put the motivation for talking about it in a broader context first, before going into the details.
So as you all know, one of the central challenges that we face in fundamental physics today is figuring out how to get rid of and replace our idea of space-time with something else. And this forced on us-- actually, as I'll discuss in the lesson today-- it's forced on us by a very, very simple combination of both quantum mechanics and gravity. Quantum mechanics and gravity together make it simply impossible to talk about any local observables.
The reason for this-- people give formal reasons for this, having to do with [INAUDIBLE] variance, which are not really the point. Because if you want this in variance as a gauge of symmetry, so it's meaningless. So whether or not something is or isn't [INAUDIBLE] is up to you. But it's actually something physical about gravity. It's really something physical about gravity and its universal attraction that gives rise to the problem.
And the basic point is this-- let's say you want to measure any local quantity. You want to measure, let's say, a 2-point function for some scalar field, or more prosaically , the temperature in this room-- the way [INAUDIBLE]-- but an experiment that takes place in a fixed region of space. Then we should ask, what do we have to do to get a sharp measurement? OK? And because of quantum mechanics, there's always two infinities involved in any sharp measurement. One of them is that we have to do the experiment infinitely often. OK?
So if you really do the experiment over, and over, and over again, you get different results every time you measure the 2-point function, because the world is quantum mechanical. But if you do the experiment often enough, the distribution becomes more and more sharply peaked around some value they you call the 2-point function. OK? So that's one infinity. You've got to do the experiment infinitely often.
Another infinity is that you have to do the experiment with an infinitely large apparatus, and that's because of decoherence. That's because, in order to come up with an arbitrarily sharp result, you have to correlate states of the apparatus with the states of the system. But there is always some exponentially tiny probability of a quantum fluctuation in the system itself, that would make the result of what you had done meaningless. If I were to do an experiment in this room, I could measure the 2-point function. If we designed a perfect experiment, I could reduce the systematic errors and the statistical errors, and get things down sharper, and sharper, and sharper.
But at some point, I can't discount the chance or the probability [INAUDIBLE] e to the minus 10 to the 30, which is e to the minus the number of things I'm made of. But I have fluctuated. I turned into a ball of dust, or that I wrote down the wrong answer, because my brain fluctuated. OK? So that means that any sharp result-- because of quantum mechanics-- involves these two infinities. We have to do the experiment infinitely often, and we have to do it with infinitely large apparatuses.
And now, gravity makes one or both of those infinities impossible. OK? For example, it makes the infinity of making the apparatus painfully large impossible, because making the apparatus infinitely large would not necessarily make it infinitely massive, if you want to keep the measurement to the confines of this room. And that means that if I want to do a measurement-- any local measurement at all-- I simply can't reduce the systematic errors down to 0. OK? There's nothing I can do to reduce the systematic error down to 0. As I start making the apparatus bigger and bigger, at some point there's so much mass in the room that I collapse the room into a black hole, and I can't make the measurement anymore.
So when we say there's no local observables, and our experimental colleagues ask us, so what the heck am I doing in my lab all the time, you idiot-- of course there's local observables. What am I doing? What's the answer to this question? The answer is, you're doing whatever the heck you're doing-- it's not perfectly sharply defined. There's no experiment you can possibly do that will reduce the systematic error on the quantity you're measuring to 0.
And if you're doing the experiment in a room which has some size r, then because of gravity, there's an inevitable error. You cannot reduce the systematic error beyond e to the negative of the entropy of the black hole that would eat up the room. OK, o e to the negative r squared over g Newton, e to the negative m Planck squared r squared. OK?
So that's just a fact. There is no local observables, because of dynamical gravity. You see, it has nothing to do with [INAUDIBLE]. If I sent [INAUDIBLE] to infinity, the theory would be just as [INAUDIBLE] as it was before, but this would not be a problem. There are local observables. OK? It's gravity that makes it impossible to have local observables.
So what do we do? We retreat to boundaries. OK? We retreat to the boundaries of space-time, where we can make the apparatuses infinitely bit. We can do the experiment infinitely often. And then the sorts of observables that we talked about are boundary of [INAUDIBLE]. OK?
So most famously, in the [INAUDIBLE] correspondence, we talk about boundary correlation functions. And so this is a startling fact. There is no inside of space-time. OK? So there's no observable associated with local measurements. It must be that the inside of the space-time is somehow an emergent concept, emerging out of something else. And so it's a very dramatic fact that we need to get some handle on.
You might naively think that all of these effects are exponentially small, until you get down to the Planck length, which is true. The most obvious effects are exponentially small. It's possible for some of these exponentially small effects to become amplified, if you do sufficiently refined measurements. So if you do [INAUDIBLE] e to the m Planck squared r squared measurements, or in any other way-- get the large, either the plus s number involved somewhere-- then it's possible that these tiny errors actually accumulate, and give you a very surprising deviation from standard expectations.
That's what, for example, happens in the black hole information problem. You can only decide whether information has or hasn't been lost if you do measurements on at least s over 2 of the [INAUDIBLE] that come out of the black hole, where s is the black hole entropy. You can't even tell the difference. The issue doesn't even arise until you do that many measurements.
But you see, your antennas should now be pricked from wondering that something might be wrong, because you're doing an incredibly large number of measurements-- incredibly exclusive questions you're asking. So there is a potential for these tiny violations of locality to pile up and give you an order one deviation from what you expect from long-distance physics.
Nonetheless, these are all qualitative arguments that tell us the inside of the space-time doesn't exist. Nothing beats having a working example of how that comes about. And the working example that we have is AdS/CFT. As you all know-- you've all heard these slogans-- AdS/CFT is wonderful. But the one thing it doesn't tell us is how to think about emergent space-time, with the emphasis on time. And the reason is-- and this is largely why it was possible for us to discover it, as incredible and mind-blowing a thing as it is-- it was so accessible to humanity because, in this particular case, the time that lives on the boundary is the same as the time that lives in the bulk.
So the theory that you're talking about-- this dual theory-- still has time. It's still a normal quantum mechanical theory. In fact, it's so normal, it's a normal particle theory. OK? So it's a gateway drug. AdS/CFT is a gateway drug into this idea that space-time is emergent. But it doesn't force you to give everything up. In particular, it lets you think about-- it's still a totally normal theory with ordinary time.
However, as we've understood more and more things, it's clear that the real situation that we have to deal with in our actual universe is very, very different. And in fact, the very, very hardest set of questions have to do with cosmology. The very hardest set of those have to do with making some sense of eternal inflation. And the reason there is that there is even less for us to talk about. There's even less for us to precisely talk about.
Even before talking about what the theory is, or what the measure is, or any of these questions, it's important to ask, what could possibly be precisely measured? OK? What could possibly be the output of some experiment that will give a totally sharp answer, for which a putative theory would predict that number? See, theories in physics don't predict fuzz. They predict sharp numbers. And we have to figure out what those sharp numbers correspond to. What observables do those sharp numbers correspond to?
And the difficulty in cosmology is that often, that second infinity of doing the experiment infinitely often gets removed from us. OK? And so now, sometimes in cosmology, there's a little bit left. If we're talking about an ordinary situation, where we have an expanding universe-- there's no cosmological constant. It just expands out into flat space. OK, there's one last observable left us which, as we might imagine, sitting on our back, and looking out at the sky, and counting how many blue stars are there, how many red stars are there, how many green stars are there-- making a big table of all of those guys-- and we get infinitely many blue stars, and red stars, and green stars. And we can measure the light with infinitely big apparatuses, because the universe is becoming infinitely big.
And so that's a sharp number that we can predict-- the fraction of red stars, and blue stars, and green stars. It's not much, OK? It's not much, but that's life. Our life in physics for the past 100 years has been-- in the past 120 years-- has been learning to deal with being able to predict fewer, and fewer, and fewer things. OK? But you think about-- this is nothing compared to what was possible at the quantum. Laplace would have committed suicide if he knew what was coming in the 1900s. OK? It's nothing compared to that.
Nonetheless, OK, there's something left. There is something left. You can talk about the fraction of red, blue, green stars. That's otherwise known as the wave function of the universe. OK? So there is a sensible notion of the wave function of the universe, which, in an infinitely large universe, tells you the relative frequencies of blue stars, green stars, and so on.
OK, so that's the last gasp. So we had boundary correlation functions in [INAUDIBLE] space. In the other extreme, in cosmology, we maybe have this fractional of blue, green, and red stars. Or if we have an accelerated universe, or eternal inflation, or many other things, it's not obvious what we have at all. OK? There's not clearly anything left that we can, in principle, measure with arbitrary precision.
And there have been various attempts in the context of eternal inflation-- if we tunnel out into flat space-- to associate some perfectly sharp, observables associated with the guys that live in the regions that tunnel out into flat space. But it's still not clear, after all this effort. Ignoring whether or not there's a dual CFT, or this, or that, it's not even clear there are such observables associated with those people. OK?
So again, before talking about the theories, we should figure out what the theories are supposed to talk about. And it's not clear at all that there is anything there. That's, I think, the biggest conceptual challenge in our kind of physics today, which is ultimately going back to the problem-- how do you do quantum mechanics, where there isn't the good separation between observer and system?
And the standard way of talking about quantum mechanics necessitates not only a separation, but the invocation of infinite systems and infinite measurements. And those infinities are idealizations that seem to be removed from us, in the context of cosmology. That's seems like a very hard problem. And it's possible to try to make some frontal assaults on those problems, and several people have done so, and continue to do so. And you're going to see how this is getting related to scattering amplitudes in a second.
This is how I got into thinking about scattering amplitudes. It was via exactly these sorts of-- thinking about these sorts of questions. But realizing that there's clearly-- I think, some genuinely new ideas are needed to make sense of these very, very hard problems. But it also seems like a very, very large leap from where we are right now, especially since there is a set of problems just one step harder than the ones that we figured out already, that we still don't understand very well. But it seems quite clear that understanding them will definitely provide us framework to thinking about everything else properly, because understanding this set of problems is going to force us to figure out where time comes from as well-- to understand not just the emergence of space, but the emergence of space-time.
And all of the difficulties with cosmology have to do with the fact that the boundaries we're talking about-- the places where observables might be defined, maybe they're not defined-- they all have to do with the fact that there is evolution in time. So we clearly need to understand where time comes from, and we have to understand it in a much more controlled setting than these infinitely hard problems, having to do with cosmology.
Fortunately, we can, because the next set of problems after [INAUDIBLE] are to understand flat space. OK? Before going on to positive cosmological constants and everything else, it seems to make sense to understand fully the physics of flat space. The observables in flat space, which were understood long before the observables associated with AdS are scattering amplitudes.
By the way, the whole argument that the only observables in gravity are associated with the boundary-- all of these things, essentially holographic ideas. Right? Let's say that only the boundary makes sense. These are not modern ideas. These ideas go back to DeWitt in the 1960s, who understood these things perfectly, and was therefore motivated to start understanding scattering amplitudes. He, in fact, did some of the first computations of scattering amplitudes-- both in gravity and in gauge theory-- and started seeing some of those surprises that, well, that have exponentiated since then.
But anyway, so the observable in flat space is the S-matrix. OK, and so what we'd like to do is see if we can understand the S-matrix. Now, you might say, well, we perfectly well know how to talk about the S-matrix, let's say from for perturbative theories-- gauge theories of gravity and so on. Feynman solved this problem a long time ago. You just draw lots and lots of Feynman diagrams, and you add them all up-- done. OK?
Where are all these tricky things coming into play? These tricky things, maybe, come into play if you're talking about making a black hole-- near the singularity of a black hole. Maybe something funny is going on there. But surely, if you're just talking about perturbative gauge theory, or perturbative gravity scattering amplitudes. There is no better way of computing them than Feynman diagrams. There is no dual theory for that, right?
AUDIENCE: What do you mean by perturbative gravity?
NIMA ARKANI-HAMED: Oh, well, you just compute low energy gravity scattering amplitudes. You can [INAUDIBLE] compute tree amplitudes, maybe some loop amplitudes. Maybe there are--
AUDIENCE: [INAUDIBLE] problem, [INAUDIBLE] of course.
NIMA ARKANI-HAMED: No, but the loop is a problem, and maybe there's counter terms you need to start talking about more and more. Maybe I just started including higher dimension operators, and systematically figure out how they affect the low energy theory. Still, there is such a thing as perturbative low-energy bounding amplitudes-- even the whole theory of string theory, whatever. There is still the question of figuring out how to compute the scattering amplitudes.
So you might think-- if you want to study these questions about gravity, ask the questions where they become important. Ask the questions where these effects become [INAUDIBLE]. These are questions about black holes. These are questions about the big bang singularity. There are questions like that. Don't ask-- it's not going to be interesting to ask the question anywhere else.
And I think this attitude turns out to-- and if we believe that, then there's many wonderful theoretical puzzles, and lots and lots of constraints, and so on. And this has been a great activity that has been going on for many years, and will go on for many years. But there is no data to guide us. Right? Because we have to go to the Planck scale to see any of these things.
Fortunately, I think this attitude is basically incorrect, and that, in fact, there is a very powerful imprint-- very powerful imprint-- of the underlying theory, without space-time, assembled on dual descriptions that we're looking for now, for flat space. There is a way of looking for it-- in fact, clues for what it is, and the sort of shape it should take-- already sitting there, in the structure of ordinary scattering amplitudes for ordinary theories, even without gravity-- even gauge theories, even the dynamics of gauge theories. The scattering amplitudes for gauge theories have clues in them that tell us, and directly address these questions about the emergence of space-time.
This may sound crazy at first. But in fact, it's part of a very long-standing tradition in physics. Just to give you an analogy, [INAUDIBLE]-- I'll just go on for a few more minutes, and then I'll start getting to the meat of the matter. But imagine that you're a classical physicist in 1850. OK? You believe in Laplacian determinism, Newton's law is everything. OK? But someone came and told you that they have a clue from the future. OK?
Determinism is gone. Funny things happen at tiny scales in phase space. All right? That's it. That's all they tell you. It's the Ghost of [INAUDIBLE] Future. OK? It comes and just gives us hints, and goes away. What would you do with this information? Well, it looks astonishing. Right? Because you say, how can determinism possibly be gone? Determinism is-- it's hardwired into our laws of nature. You know? I can't take Newton's laws and just tinker them a little bit, and make them a little bit non-deterministic. OK?
What the heck could this other theory possibly be, that pulls the rug out from us so dramatically? And if you thought about it some more-- maybe I should make the year 1780, just to be more historically accurate. If you thought about it some more, you might say, well, that's just nuts. There is no way, because whatever that other theory is, it has some parameter. There's some limit. There's some correspondence limit. Somehow, it has to smoothly connect to our theory.
[INAUDIBLE] isn't wrong. It's damn good. Right? What possible small perturbation could we make to our theory that makes it, all of a sudden, not deterministic? OK? This person isn't coming and telling me determinism is a little bit off. They said there's no determinism, fundamentally, at all. It's just gone. Right?
How could that possibly be incorporated into your picture of the world? Well, you might think, maybe what I should do is think about what funny things could happen at small areas in phase space, OK? Because that's another clue that you were given. And maybe you think for a while, and you have the brilliant idea-- you put phase space on a lattice. OK? Oh, wow. Phase space on a lattice-- that's just awesome.
Then you think a little more, and you realize that's actually tough with theorem. OK? There's diffeomorphism in variance in phase space, so just putting it on the lattice breaks it. This analogy should be of interest to people who think about the quantum gravity. OK? We have almost the same words today, with-- anyway.
That's totally wrong. You might take that as a clue, but it's not the right way to go. You might, with some work, come up with semi-classical ideas, like the Bohr-Sommerfeld quantization rules, or something like that, and get some inkling of what that theory is like. But still, you're not seeing this 0-th order big perturbation, that determinism is gone.
But there is another thing you could do, which would not require you to think about what's going on at tiny distances in phase space, and modify the theory you have. You could ask-- the fact that there is this other new theory, where determinism is gone, and the fact that it needs some correspondence limit that takes us back to our theory, must mean that there is some way of formulating our current theory, where determinism is not manifest. That's the only possible way there could be a correspondence limit.
And you would think, and you would think. And you would invent the least-action principle. OK? The least-action principle is just a reformulation of classical physics. It's nothing new. It's just a reformulation of classical physics. But it's a reformulation with a very, very important feature. Determinism is not manifest in the least-action principle. In fact, when you first learn it, it's very [INAUDIBLE]. You say the particle goes from A to B, as if it sniffs out all possible paths and takes the path of least action.
AUDIENCE: [INAUDIBLE]?
NIMA ARKANI-HAMED: Well of course, it ends up being a complete equivalent to Newton's laws. But in the formulation, determinism is not manifest. Determinism is, instead, an output. OK? You discover, given the rules for how you do the variation, that, oh-- with these rules for doing the variation, the actual-- it looks like local equations of motion, which are deterministic. But the basic laws is not deterministic. The statement is not deterministic. OK?
Determinism is an emergent concept, from the point of view of the principle of least action. Other things become obvious-- the symmetries, the link with conservation laws. All sorts of other things become manifest. But determinism is not manifest.
In fact, this is not how things happened, because there was no Ghost of Theorist Future. The way things happened is that people started doing harder and harder calculations in classical mechanics, and they found that Newton's laws were increasingly cumbersome. And they invented the principle of least action as a wonderful reformulation of the theory that there' a lot of the calculations that become simpler. OK?
And it did bother a few of them that, sitting there, was this completely different outlook on what the world looks like. Right? Determinism does not seem to be manifest, but it's just a re-writing. There's nothing new. There's no hbar, nothing. Right?
AUDIENCE: So those people really [INAUDIBLE]?
NIMA ARKANI-HAMED: No, absolutely. In fact, people have said-- well, I can tell you about the history a lot later. But people talked of the principle of least action as a proof of the divine, right-- or telling you that there is some pre-ordained way things are, and that everything is as perfect as can be, and so on. OK? So anyway, the history is fascinating, but I'll probably tell you more about it later.
So the message here is that it's this playing around with the theories you have, and finding other ways of formulating them will make different features manifest has had a long and glorious tradition in our subject. And in fact, the point is that different formulations can be of different degrees of utility in making contact with the next theory up. OK?
Newton's laws are a terrible way of making the leap to quantum mechanics. The principle of least action, which is just a rewriting of Newton's laws-- is a wonderful way of making the leap to quantum mechanics, because after playing around with it for a while, you might start wondering, what are all those other paths for? Do they have any point in life? OK? And eventually, you might come up with-- this is perhaps a little far-fetched, but you might come up with the idea of quantum mechanics, as a natural deformation of that theory that you have. OK?
So once again, it's not quantum mechanics. It turned out to be useful for predicting orbits of planets and all sorts of things. There's no hbar in it. But in a sense, it did almost all of the job for going to quantum mechanics. OK? It figured out some way of talking about completely deterministic physics, [INAUDIBLE] didn't put determinism in by hand.
That's the biggest trick. Right? After that, it's a little modification from there to allow the possibility that there's determinism there at all. OK? But most of the trick was done-- or a large part of the psychological hurdle was overcome-- by realizing that there was this second way of talking about things when determinism [INAUDIBLE]. OK.
I claim that that's exactly the situation we're in today, with the word determinism replace placed by the word locality. So we have quantum mechanics. We have quantum field theory. But we know-- in this case, we have the Ghost of Theorist Future. The Ghost of Theorist Future is the thought experiments that tell us that local observables don't exist because of quantum mechanics and gravity. So we know that, as a clue, for what the correct formulation of physics looks like.
Then the question is, what do we do? And a perfectly defensible, important set of things to do is to wonder what happens down Laplacian distances. But another thing that we could be doing is to try to reformulate the theories we know, and love, and have, in a way that eviscerates space-time-- takes it out completely, but gets the observable results back. It gives us the scattering amplitudes directly, without talking about an evolution through the area of space-time. So that's a motivation for studying scattering amplitudes for ordinary theories, even without gravity, OK-- just ordinary gauge theories that we know and love.
Now, that was a long and highfalutin set of arguments. Let me say one last thing about it, though. Unlike trying to figure out what's going on in the Planck scale using thought experiments or other things, this direction has the advantage of having an enormous amount of data. OK? It's a gigantic amount of data. Every scattering amplitude that's ever been computed is data. OK? Because it has to be reproduced by a putative dual theory. OK? And you know when you're on the right track or you're not, because you're either computing them or you're not. There's no bullshit in this business-- zero, OK? It either works or it doesn't.
OK. So those are very, very highfalutin arguments for why there should be some way-- some formulation of standard scattering amplitudes for gauge theories, where you don't put in space-time by hand, but space-time is an emergent concept. And perhaps other things are made more manifest. However, there's a much, much more pragmatic set of arguments, which remarkably line up in exactly the same direction. In fact, this is really the analog of people inventing the principle of least action to do the hard calculations. OK?
So here, you actually start trying to do gauge theory calculations of scattering amplitudes for anything other than two-to-two processes, and you run into the famous fact that Feynman diagrams suck. OK? They just suck. So if you want to compute-- and by the way, Feynman himself knew this, and complained to anyone who would listen that his methods were no good for many, many extra particles. OK?
But let's talk about a process like two [INAUDIBLE] to two [INAUDIBLE], going to four [INAUDIBLE]-- a process that is an important background to any search you're interested in at the LHC, with even relevent-- can happen hundreds of thousands of times a second-- very, very important process. So if you count your 220 [INAUDIBLE] diagrams, there is tens of thousands of terms, [INAUDIBLE] it's just totally ridiculous. There are tens of thousands of terms.
And people had to-- and in fact, [INAUDIBLE], amongst other people, had to come up with tricks for computing them. It's hopeless to sum the [INAUDIBLE] diagrams. So you come up with tricks. At first, it looks like tricks, OK-- shortcuts to doing the calculations and so on. And what we find is extraordinary-- that the some of these tens of thousands of terms are captured in either one term or three terms, total.
AUDIENCE: [INAUDIBLE].
NIMA ARKANI-HAMED: Well, it depends. [INAUDIBLE] for example, I'll give you the answer. If all the [INAUDIBLE] are plus, the amplitude is 0. If all the [INAUDIBLE]-- so here we're imagining all the [INAUDIBLE]. If all [INAUDIBLE] are possible, [? what it will ?] remind us is the amplitude is also 0. If two of the [INAUDIBLE] are minus, let's say for particles i and j-- and I'll review what the [INAUDIBLE] means in a second-- then the amplitude is this. But for now, you're just supposed to be impressed by the fact that it's one term-- just a single term.
And the most [INAUDIBLE] configuration is with three pluses and three minuses. And I won't write it down, but it's three terms. OK? They're a little bit more complicated than that, but there's three terms. OK? This is just stunning. It's absolutely stunning.
So [INAUDIBLE] diagrams are clearly hiding something. They're hiding a simplicity to the answer. OK? Now, why are [INAUDIBLE] diagrams complicated? [INAUDIBLE] diagrams are complicated because they have one job to do in life. They make locality and unitarity manifest. That's the whole point. OK? You've got [INAUDIBLE], which comes from [INAUDIBLE]. Locality and unitarity are the things that are being made manifest by [INAUDIBLE] diagrams.
If you have something like a [INAUDIBLE] theory, or a [INAUDIBLE] theory, or a crappy theory like that, it turns out-- I'll just tell you now-- there is no [? extra ?] simplicity to the answer, compared to what you'd expect from [INAUDIBLE] diagrams. What you get is about as bad as what you'd expect. There's no [INAUDIBLE] simplicity.
The [INAUDIBLE] simplicity starts for gauge theories, where the [INAUDIBLE] diagrams are much more complicated than for [INAUDIBLE] or [INAUDIBLE] theory. But the answers turn out to be much simpler than [INAUDIBLE] theory. And remarkably, [INAUDIBLE]-- I won't talk about it a lot in these lectures-- for gravity, in a specific sense, they are simpler still, even though the [INAUDIBLE] diagrams are now a complete nightmare.
There's 80 terms in the cubic vertex for gravity [INAUDIBLE]. It's just a total nightmare to do the calculations directly, but the answers are simpler-- even simpler than in gauge theory, in this specific sense. OK? So already, there is something really bizarre going on. There is a complete inversion from what you think is simple and not complicated, and the actual answer. Forget about the process of getting the answer. The actual answer doesn't betray any of this fact. You wouldn't-- it leads you to expect the opposite from a [INAUDIBLE] diagram.
But why is it? Where is this complexity in [INAUDIBLE] diagrams coming from, and why isn't there in gauge theories and in gravity? And the reason, once again, has to do with the fact that it's making locality and unitarity manifest. It does that. So what is an actual scattering amplitude? An actual scattering amplitude in four dimensions is an object that's labeled by-- it'll give you a bunch of strings with plus 1's and minus 1's, with [INAUDIBLE] particles and [INAUDIBLE] particles.
And this deserves to be called an amplitude if what you do [INAUDIBLE] you get a [INAUDIBLE] transformation on the [INAUDIBLE]. OK? So that's what a scattering amplitude is. We'll always imagine [INAUDIBLE] are incoming. OK?
But that's not what you compute with [INAUDIBLE] diagrams. What you compute with [INAUDIBLE] diagrams are things that have [INAUDIBLE] indices. And to get the actual amplitude, you [? dot ?] these indices in their polarization vectors. OK? And then you hope that it's true-- and you say that it's true-- that these polarization vectors are like [INAUDIBLE] between the [INAUDIBLE] group and the [? little ?] group. OK? They have a property that if you do a [INAUDIBLE] transformation on them-- on their [INAUDIBLE] index-- you pick up a little group rotation on their [INAUDIBLE] index. OK?
You pretend that such an object exists. [INAUDIBLE] amplitude, and you get the answer. You [INAUDIBLE] and you go home. OK?
AUDIENCE: But what if [INAUDIBLE]?
NIMA ARKANI-HAMED: No, it's pretend because those objects don't exist. There is no such thing as a polarization vector. There is no such thing as a polarization vector that behaves as a [INAUDIBLE], where if you do a [INAUDIBLE] transformation on it, you get a [INAUDIBLE] rotation on the index. It doesn't exist.
The difficulty has to do with the famous fact that there's only two polarizations, let's say, of a [INAUDIBLE] particle, or any massless [INAUDIBLE] particle. And what actually happens is [INAUDIBLE] do under-- if you're having epsilon [INAUDIBLE] p, it can transform into [INAUDIBLE] epsilon [INAUDIBLE], but there's always something proportional to [INAUDIBLE] that you can't get rid of.
That's the basic problem-- that there's two [INAUDIBLE]. And there is simply no [INAUDIBLE] way of picking out two out of four [INAUDIBLE]. Because I don't use [INAUDIBLE] to make locality manifest, I have to have fields with [INAUDIBLE] indices, and that's why the [INAUDIBLE] of our [INAUDIBLE]. But I have four degrees of freedom here. I can try to knock it down by placing [INAUDIBLE] constraints on it like that, which knocked it down to three, but that's it. There's nothing else I can do. There are still three degrees of freedom-- not two, three.
And very concretely, what that means is, even if you say, OK. Screw all this [INAUDIBLE] crap. I'm going to just pick a frame and say, you're my [INAUDIBLE] vector. OK? When the particle's moving in the z direction-- so [INAUDIBLE] was this, that's [INAUDIBLE] correlation vectors. Don't give me a hard time. Go home. OK?
That's fine. You're perfectly free to do that in [INAUDIBLE]. But in another [INAUDIBLE] frame, it will not look like that. That's the problem. In another [INAUDIBLE] frame, the corresponding epsilon will have a component proportional [INAUDIBLE]. There's simply no way of getting rid of that component that's proportional to [INAUDIBLE].
And so what we have to do is identify-- this is really a new thing. We have to identify epsilon [INAUDIBLE] and epsilon [INAUDIBLE] plus anything times [INAUDIBLE], and declare that they're the same physical state. That's a declaration. OK? So the polarization vectors are not-- there's no unique polarization vectors. There's a whole family of polarization vectors, all corresponding to the same physical state, differing by amounts proportional to [INAUDIBLE].
Of course, this is nothing other than the [INAUDIBLE] space version of [INAUDIBLE] variants. OK? We have to demand that these-- this is a redundancy. We have to demand that these give us the same physical state. So [INAUDIBLE] and [INAUDIBLE], if it's coming from a [INAUDIBLE], the only way you have a prayer for this to be true-- and this means something very specific. Correct? This means that these [INAUDIBLE] that we calculated have to satisfy that if you [INAUDIBLE] any of them, you have to get 0. OK?
Because it can't know which representative you chose. But depending on which representative you choose, they differ by [INAUDIBLE]. The only way the physical amplitudes are going to be invariant is if this is true. But now we have a tall order ahead of you. You have to figure out how to build a theory, which is going to guarantee that this is true, when you sum all of its [INAUDIBLE] diagrams.
The only way you have a prayer in hell of doing that is if you've built all this redundancy into the theory. You build gauge redundancy into the theory, and so on. So gauge redundancy is the price we pay for making locality and unitarity as manifest as possible.
Let me say it another way. We have the [INAUDIBLE] effect. And we say, oh, it looks non-local, but don't worry about it. It's really local. Why do we say it's really local? Because we know, in our heads, we can introduce a vector potential, and it [? couples ?] [? lovingly, ?] and everything is fine. But that's at the price of a great deal of redundancy. OK? We can be non-redundant and give up manifest locality, or redundant and keep locality.
OK. So that's the reason for all the complexity in the [INAUDIBLE] diagrams. We're carrying all this gauge redundant baggage about--
AUDIENCE: So is it related to the [INAUDIBLE] vector [INAUDIBLE] vector field without gauge symmetry?
NIMA ARKANI-HAMED: Yes.
AUDIENCE: We cannot just say, oh--
NIMA ARKANI-HAMED: Exactly, yes. It's exactly the same thing. There are many guises of exactly the same problem. I'm choosing to emphasise the thing that highlights the physical observable the whole way. Because we're not-- in a moment, we won't be talking about [INAUDIBLE] ever again. OK? But I want to say, if you want to describe the [INAUDIBLE] or massless particles, a crucial point here is the massless particles.
I should say, where did unitarity come in? Unitarity came in because it's unitary representations of the [INAUDIBLE] group for massless particles that tell you that there's only [INAUDIBLE]. So it's literally these two ingredients-- unitarity and locality, boom-- that the only way of making them manifest is to build all this gauge redundancy. And the gauge redundancy horrendously complicates the calculation.
And furthermore, it all cancels out in the final answer, which is how the 220 terms go to 1 term or 3 terms, even at tree level. But you can't even isolate-- oh, that term canceled that term. It's horribly non-locally spread out, throughout the calculation. It just happens, right at the end, it collapses to these extremely simple, simple results.
AUDIENCE: So in this picture, if I would assume [INAUDIBLE] gauge [INAUDIBLE] and just tried to [INAUDIBLE].
NIMA ARKANI-HAMED: Yes, yes.
AUDIENCE: Where are you going to see the problem?
NIMA ARKANI-HAMED: Well, if you just [INAUDIBLE] the vector field, and you're just an idiot, and you write [INAUDIBLE], [INAUDIBLE] all sorts of problems. You have negative [INAUDIBLE] states. You have all sorts of other problems. OK? Then you could start from that and try to clean up the mess that you made. That's one of the standard approaches to what we're talking about. Most field theory text books take that out of it.
From this point of view, that's all backwards, because the actual observable is the scattering amplitude. And you're trying to build a [INAUDIBLE] to make locality and unitarity manifest. It has a job to do. That's the job it has to do. OK? It has to ensure that these [INAUDIBLE] are satisfied. And then you just have to go find the [INAUDIBLE], which makes that happen. And this gives you a good clue that it should have some gauge redundancy in it, because it needs to know that these two polarization vectors [INAUDIBLE].
So this is the situation for gauge theory. For gravity, we have even more redundancy. OK? That's the [INAUDIBLE]. And so normally-- so all of this gauge redundancy is there. The [INAUDIBLE] locality and unitarity manifest. But it comes at a big price. It makes the structure of the answer-- the simplicity of the answer-- it completely obscures it. OK?
So this is another very basic fact that I find-- and people who get into this business find-- absolutely mesmerizing, is that there are these all highfaluting arguments, and oh, space-time, gravity, all of that. But sitting under our noses for 60 years, in the structure of good, old-fashioned scattering amplitudes and QED, QCD-- normal theories-- there's been this big clue. There's something wrong with space-time. There's something wrong with taking these ideas so seriously, because when you do, you force these horrendous complexities in the answer that are not there-- in the process of computing the answer, that are not there in the answer itself.
There must be some way of getting this answer in another way, which doesn't put space-time in, OK-- and maybe not unitarity as well. So because all the problems have to do with putting the two of them together. Again, it will just be a dead reformulation. So it will be completely unitary, completely local. But those properties will be emergent, and some other things will become obvious. OK.
A final note-- this fact, that the polarization vectors don't really exist, but are only a member of a redundant class, is the reason why most of us-- in our field theory classes-- never saw a scattering amplitude, actually, because it was always given in that form. So we never actually saw it. We always saw some redundant representative of it, but we never saw what the amplitude itself was. OK?
We'll talk in a moment about the variables to use-- which exist, very thankfully, in four dimensions-- that let you see directly what the amplitudes actually are, as functions. It's the beginning of understanding the underlying structure. But let me just say a few words about the development of the subject, and where it is now, before getting into details.
AUDIENCE: Can I ask about the [INAUDIBLE] case first?
NIMA ARKANI-HAMED: Yes.
AUDIENCE: So you're going to use this redundancy, which would get rid of the redundancy in the gauge field case or the gravity case.
NIMA ARKANI-HAMED: Right.
AUDIENCE: You're going to recover some much simpler formulation, which allows you to argue that we're seeing space-time emerge from a reformulation of quantum field theory.
NIMA ARKANI-HAMED: Exactly.
AUDIENCE: The fact that you apparently don't get much of a bonus by reformulating [INAUDIBLE] theories is telling you what exactly?
NIMA ARKANI-HAMED: I think it's telling us that [INAUDIBLE] field theories are not actually-- are not a fundamental part of nature. And you think of them as bits and pieces of underlying theories [INAUDIBLE] gravity. As we know, they are. So there are [INAUDIBLE] interactions in [INAUDIBLE] theory, or in [INAUDIBLE] theory. And so there are [? scalars ?] around, and so on. But that's really fundamentally what it's telling us.
And of course, we know that. Right? We know that, even all by ourselves, because we know that there is something wrong with just the [? scalar ?] theories. They have hierarchy problems. They don't want to survive [INAUDIBLE], and so on. So [INAUDIBLE] find that as yet another little push, that this line of thinking is actually correct. You see, this is a-- yes?
AUDIENCE: I was just going to ask a historic question, but [INAUDIBLE]. So just wondering whether Roger Penrose's [INAUDIBLE] was [INAUDIBLE].
NIMA ARKANI-HAMED: Yes, [INAUDIBLE] are going to play a very big role in this story. And I should say that Penrose understood many of these things about the importance of massless scattering amplitudes, the importance of looking at the boundary and so on, really, with amazing foresight, already in the mid '60s. So DeWitt understood these things. Penrose understood them very, very well.
So the philosophy-- this basic holographic philosophy, these guys understood cold, OK, already in the 1960s. And it's an absolutely wonderful part of the story that these ideas that Penrose and his group had, in Oxford, been developing for 30 years, finally, finally have found a real home. And they are really-- by the end of the lecture, I hope you see that you will never think about scattering amplitudes without using twisters ever again. OK? It's just insane any other way. But it took 40 years to see it.
But nor was it a case of [INAUDIBLE] gradually chipping away, until it made sense. It was really [INAUDIBLE], I mean it was a collection of things-- people doing the computations in gauge theories, getting the data, mining the data, getting the data. You start seeing the data. You get more and more of the data. If you start seeing patterns, you start suspecting them.
This is really-- this is the little bit of history I want to say. So it's remarkable that all of this was largely motivated from the grungy part-- the non-new physics part-- of Hadron collider experiments, which is to compute the [INAUDIBLE]. OK? So this is why people did these calculations. It's a truly wonderful fact about physics that good deeds are eventually rewarded. Doing morally correct things are eventually rewarded.
I mean, people like me-- I will fully admit-- were like, oh, poor guys, doing these grungy calculations. No one gives a shit about them. They don't-- oh, well. Someone's got to do it. [INAUDIBLE].
[LAUGHTER]
No, I'm being honest. And this is complete, total-- I always say, I was a postdoc at [INAUDIBLE] in the '90s, with Lance Dixon two offices down. And had I just talked to him then, my life would have been quite, quite different. So these people were doing heroic calculations, hard calculations. Well, [INAUDIBLE] knows very well. He was one of the people who [INAUDIBLE] these difficult, very difficult, wonderful calculations.
But they were not just important for-- I mean, they were important for all the things they're actually important for. But they began to develop this big amount of data [INAUDIBLE]. But there was something going on here. These answers-- it's not just a trick here or there. Of course, you think it's tricks. Really, it's quantum field theory [INAUDIBLE] isn't field theory amazing? It gives us these incredible answers.
What explains [INAUDIBLE] so simple? Well, it's just field theory. Field theory's an amazing thing. But you start suspecting, slowly by slowly-- as more and more data accumulates, more and more bizarre connections between the answers accumulate-- that there really might be something going on. Right? And it's not just that the answers are simple, but that there really might be another formulation. OK?
Now, this program got a huge boost from [? Witten ?] in late 2003, where he suggested that there was a dual formulation of [INAUDIBLE] theory. I'll explain when we get into it. You'll see why it is that [? maximal ?] [? supersymmetric ?] theories-- you would have discovered them all by yourself, just thinking about scattering amplitudes, because they're so much nicer to think about scattering amplitudes than ordinary theories. And you'll see why.
So [? maximal ?] [? supersymmetry ?] is an incredibly natural thing, from the point of view of scattering amplitudes, and we'll be spending a lot of time talking about that. But anyway, at tree level, it doesn't make any difference. At tree level, [INAUDIBLE] scattering and [INAUDIBLE] scattering are just exactly the same, because the [INAUDIBLE] can't run in loops. At loop level, they're obviously different.
But as you'll see, the real thing about [? maximal ?] [? supersymmetry ?] is just that it makes [INAUDIBLE] talking about scattering amplitudes much simpler than pure [INAUDIBLE]. It has very little to do with cancellations and loops, and stuff like that. It's really something much more basic and hardwired into the observable, that you're talking about.
Anyway, so [? Witten ?] made a proposal for a dual theory, precisely this thing-- a dual theory, not just a trick-- a collection of tricks-- but a dual theory that would compute all of these [INAUDIBLE] that was associated with some string theory in twistor space-- some topological string theory in twistor space. This was really startling, because these words-- twistor space, dual theory, all this stuff was now coming and having something to do with good, old-fashioned [INAUDIBLE] scattering.
OK. It ended up-- well, at some point in these lectures, I will tell you the easy way of thinking about twistor string theory. It'll be one of the many things that comes out from this dual formulation that now exists, that I will be telling you about. But anyway, it gave lots of-- [INAUDIBLE] but it ended up being difficult to do calculations with the twistor string theory itself. There were reasons to suspect that it didn't work at loop level. But it inspired many developments.
And in fact, it inspired many developments for the much more effective and fast computation of tree amplitudes, in the form of new recursion relations for the tree amplitudes-- first, the so-called [INAUDIBLE] recursion relations, then the Britto-Cachazo-Feng-Witten, BCFW, recursion relations. The BCFW recursion relations, in particular, were extremely important. They're extremely important for two reasons. A, they are really, really, really efficient, so they allowed people to generate lots of data, OK-- just for tree amplitudes, in gauge theories, and in gravity. Five, six, seven, eight, nine particles-- you could start computing them and actually seeing what the answers look like [INAUDIBLE].
But secondly, while the people eventually understood the CSW rules, essentially being a very, very clever version of [INAUDIBLE]. OK? But they were still [INAUDIBLE] diagrams. They're [INAUDIBLE] diagrams where the vertices were associated with [INAUDIBLE], these [? maximal ?] [INAUDIBLE] amplitudes that I'll tell you more about. But they still look like [INAUDIBLE] diagrams.
The BCFW recursion relations finally didn't even look like [INAUDIBLE] diagrams. They didn't look like they came from [INAUDIBLE]. And these very short forms of the answer, that I told you-- there's three terms to the answer, [INAUDIBLE] amplitudes is when you compute it from BCFW. And they have the following remarkable property, that if someone hands you an amplitude, you can ask-- and you didn't know it came from a [INAUDIBLE]. Someone just handed you an object, saying, this is an amplitude. You could ask-- how could you tell whether it's consistent with any kind of local space-time description or not?
You can tell by seeing the kinds of poles that this object has. OK? Because if I have the external particles [INAUDIBLE], then the [? allowed ?] kind of poles that we can have, we know correspond to long-distance propagation in space-time. And that means that the amplitude should have poles that look like 1 over some sum of the subset of the [INAUDIBLE] squared.
And furthermore, the residue on that pole should factorize into some tree amplitude on the left times some tree amplitude on the right. So locality is sitting there in plain sight, in the singularity structure of the amplitude. This is even at tree level, OK-- just [? even ?] there, at the tree level.
What was remarkable about the BCFW form of the amplitudes is that they gave the answer in the terms of a few terms, some of it [INAUDIBLE] terms. But each term, individually, did not have a local interpretation. Each term, individually, had some poles that were local and some poles that were not local.
The poles that were not local, just to give you an example, [INAUDIBLE] some, let's say if we call these sort of sums [INAUDIBLE] capital P, the [INAUDIBLE] singularities are allowed to have a 1 over P squared. But for example, not 1 over P squared, q squared, minus r squared P squared. OK? That's a [INAUDIBLE]. And each individual BCFW term is riddled with those. So there's local poles, there's non-local poles.
Of course, miraculously, when you sum the terms together-- the three terms together-- you find these non-local poles cancel, and you're only left with the local ones. OK? But somehow, you really-- you see even more vividly that you're gaining simplicity at the expense of locality. OK? You're gaining simplicity at the expense of allowing objects that could never possibly come from an underlying local theory. OK? So that one was a very, very striking-- that was a very striking feature of these BCFW [INAUDIBLE].
Now, all of this--
AUDIENCE: This was a three-level argument?
NIMA ARKANI-HAMED: Sorry?
AUDIENCE: This was a three-level argument?
NIMA ARKANI-HAMED: This was just at three levels, yeah. [INAUDIBLE].
AUDIENCE: Because we know [INAUDIBLE].
NIMA ARKANI-HAMED: Sorry?
AUDIENCE: Once we include [INAUDIBLE] in the propagator [INAUDIBLE], then the [INAUDIBLE] gets shifted.
NIMA ARKANI-HAMED: Of course, the-- and we have [INAUDIBLE] and so on. But let's just talk about the-- just the tree level. That was just a tree-level statement. OK. So now, yet another thing that happened in a completely parallel development is that people started studying-- people had been studying planar [INAUDIBLE]. OK? So [INAUDIBLE] limit. And noticed that a [INAUDIBLE] curiosity-- that in just looking at the multi-loop structure, the kinds of integrals that occurred at multi-loop level had a strange feature.
They had a new kind of symmetry. They had a new sort of conformal symmetry that they called dual conformal symmetry. I'll tell you what it is in a second. It'll be hard to describe at the moment, exactly how they saw it. But I'll tell you what it is in a second, but I want to say that it was first noticed by people doing perturbative calculations, just seeing that the multi-loop integrals had a funny symmetry-- that was [INAUDIBLE].
This was really correctly and vividly understood by [INAUDIBLE], who started looking at amplitudes [INAUDIBLE] coupling, using [INAUDIBLE], and seeing that [INAUDIBLE] it was natural to introduce a sort of [INAUDIBLE] duality that mapped to the calculation of scattering amplitudes-- just [INAUDIBLE] scattering amplitudes at strong coupling to the computation of a certain expectation value of the Wilson loop at strong coupling-- but a Wilson loop whose edges were null.
So let me describe now what this dual conformal symmetry is. That was the history, but let me describe what it is. If we're interested in scattering amplitudes of-- in general, any scattering amplitudes-- we can [? pull out ?] [? the color ?] factor, out of everything, and just sum all the possible permutations of 1 though m, each one [INAUDIBLE] of its own [? color ?] factor. And if we're talking about the [INAUDIBLE], [INAUDIBLE] single trace operators. So we'll actually focus all of our attention on one of these so-called [INAUDIBLE] amplitudes, which are a function of 1 through m. OK?
And in a general [INAUDIBLE] they also have policity labels. Later, we'll talk about how to more nicely talk about it super symmetrically, and what the correct variables are. But let me, just for this purpose-- just for the purpose of introducing what dual conformal symmetry is, it suffices to notice that there's a natural cyclic structure to the [? external ?] [INAUDIBLE].
Now, momentum is conserved, so you can imagine laying the momentum of particles end to end on a four-dimensional sheet of paper. So if this is [INAUDIBLE] 1, [INAUDIBLE] 2, [INAUDIBLE] 3. [INAUDIBLE]. Then they form a closed polygon. We can make this say this formally, by saying that we can write the [INAUDIBLE] of the difference like that-- the difference between two coordinates. OK?
These are funny coordinates. They're not ordinary space-time coordinates. They're [INAUDIBLE] momentum. They're units of momentum. OK? But because there's a natural [INAUDIBLE], there is this natural polygon associated with the problem. And there is this natural dual space, x, in which the momentum-- which gives you the vertices of the polygon. OK?
You'll notice that there's some redundancy here. Obviously, I can translate everything and get exactly the same [? piece. ?] OK. What people notice is that the scattering amplitudes appropriately define-- after you do a few [INAUDIBLE] with them-- well, I'll even tell you what you have to do to them. So normally, when we have scattering amplitudes, there is a delta function of momentum conservation sitting in front of everything. OK? In a supersymmetric theory, there's a super partner as well. So there is the delta function of the super [? charge ?], the delta function of the momentum, times something. OK?
This whole picture only makes sense on the support of that delta function. OK? So [INAUDIBLE] if you take the amplitude and you strip off the trivial delta functions that conserve momentum and super momentum, then you have a resulting object that makes sense now, only on the support of those delta functions. OK? But that object is invariant under a new conformal symmetry. The conformal symmetry is conformal transformations on this space.
AUDIENCE: So the dimensionality is the same, right?
NIMA ARKANI-HAMED: Dimensionality is the same. Everything is the same, but there's a dual conformal invariance.
AUDIENCE: So do you think the [INAUDIBLE] amplitudes, so you take it away?
NIMA ARKANI-HAMED: Yeah, so if you just take away the trivial delta functions, the thing which is left is invariant, bizarrely, under conformal transformations in this space. For example, it's invariant under inversion. If you take x [INAUDIBLE] and you replace it with x [INAUDIBLE] over x squared, the amplitude is invariant-- completely crazy. You might expect this, if these x's were space-time coordinates, because [INAUDIBLE] equals 4 is conformal in space-time.
Now, at loop level there's confusions about [INAUDIBLE] should be talking about. There's [INAUDIBLE] in the amplitudes, and so on. But even if we just stick with tree level, at tree level there's a well-defined sense that those conformal transformations act on the amplitudes. That's obviously [INAUDIBLE]. Great. OK?
AUDIENCE: [INAUDIBLE]?
NIMA ARKANI-HAMED: Well, at tree level, it's also for pure [INAUDIBLE]. So it's an exact statement, apparently, only at tree level. We'll see later, when we talk about things more, it's actually a symmetry in the full loop into [INAUDIBLE], which is a sensible object to all [? loop works. ?]
AUDIENCE: [INAUDIBLE] theories at tree level?
NIMA ARKANI-HAMED: Well, no. At tree level, this is the beginning symmetry of QCD that no one ever knew about. This is what I'm telling you. These things have been lying under people's noses for-- this could have been noticed, in principle, 30 years ago. OK? No one knew to look. No one knew to look that sitting there, [INAUDIBLE] actual [INAUDIBLE] particle [INAUDIBLE] scattering amplitude was some crazy symmetry in this dual space. OK?
AUDIENCE: Does the same hold for higher dimensional [INAUDIBLE]?
NIMA ARKANI-HAMED: No. Well, yes, actually-- at tree level, it does. But look, I can tell you a little more about that later. Of course, higher dimensional theories have a problem at that loop level. But at tree level, in this specific sense, it does.
OK, so so far in this story, the main hint that there was some dual formulation was the simplicity of the answers-- simplicity, simplicity, simplicity. This, I think, just puts the final nail in the coffin of the [INAUDIBLE], because this is now an honest-to-goodness symmetry the theory has-- just as big as the symmetries that see us, in front of our face, but which are completely obscured in [INAUDIBLE].
Normally in physics, when you have symmetries, OK-- it's not aesthetics. Is it simple, not simple? It's your job to figure out how to talk about the physics so that you make all the symmetries manifest.
AUDIENCE: So if there's a [INAUDIBLE]?
NIMA ARKANI-HAMED: There are, actually. That's actually-- it's actually extremely interesting. The reason-- there's an infinite [INAUDIBLE] quantities, actually. So if you take this symmetry and the ordinary conformal symmetry, they don't commute with each other. I should say, this is like a T-duality. Right? This is really-- and this is actually where it came from, in [INAUDIBLE]. It came from an actual T-duality.
It's like a T-duality. It's a space where the coordinates are the inverse of the usual coordinates. OK? They don't commute with each other. Right? These symmetries and the other symmetries do not commute with each other. And in fact, they close into an infinite dimensional algebra. It's calling a Yangian, and there's an infinite dimensional Yangian algebra. OK? You don't even need to know what the Yangian is or what Yangian algebra is. But I'll use the word Yangian over and over again, to refer to what you get from combining both conformal and dual conformal invariance.
In other words, there's an infinite dimensional symmetry. Now, you might ask-- as [INAUDIBLE] was about to ask, I think-- doesn't this violate the [INAUDIBLE] theorem? Doesn't it violate all these things? And it doesn't, because these are extremely interesting symmetries. They are not symmetries that have well-defined action on one [? particle state. ?]
You see it here. You see, even what the symmetry is depends on [INAUDIBLE] particles. OK? So the way it acts mixes up particles with their neighbors in an interesting way. And so it doesn't have action on single states, so it doesn't violate-- it's yet another loophole in the [INAUDIBLE] theorem. But anyway, it's true.
There was an infinite dimensional symmetry, which was there in [INAUDIBLE]. Now we understand [INAUDIBLE] in the specific sense of the infinite loop [? work. ?] OK? But it's certainly there at tree level, even for [INAUDIBLE]. But you write down [INAUDIBLE], where is this? OK? It isn't there. Negative [INAUDIBLE] is letting you see half of the symmetry-- in a sense, 1 over infinity of the symmetry-- but one half of it. And the other half is [INAUDIBLE] totally obscure. OK? Yeah?
AUDIENCE: [INAUDIBLE] application, is there [INAUDIBLE] the symmetry to compute amplitudes by just computing one quantity, and then just a few quantities, and then actually using this to get the whole [INAUDIBLE]?
NIMA ARKANI-HAMED: Roughly speaking, very roughly speaking, that's what we're going to be doing. OK? So very roughly speaking, that is exactly what we're going to do. [INAUDIBLE] historically here, just to say that when this happens, you're now just convinced there must be a second way. Because again, what's making it impossible to see this is your adherence to [? local ?] space-time.
If you're picking the original space-time [INAUDIBLE] other conformal symmetry [? manifest. ?] You can pick the other one. You're completing a Wilson loop on that side. Why the heck [INAUDIBLE] scattering amplitude in the other space? God knows, OK? You can commit to one space or the other, and the commitment to any local description obscures the [INAUDIBLE] dimensional symmetry [INAUDIBLE]. OK.
AUDIENCE: So again, what is this-- [INAUDIBLE]?
NIMA ARKANI-HAMED: Dual.
[INTERPOSING VOICES]
NIMA ARKANI-HAMED: The Yangian, Yangian-- the Yangian [INAUDIBLE].
AUDIENCE: [INAUDIBLE].
NIMA ARKANI-HAMED: So that-- so I can explain that. I can tell you, but it's utterly irrelevant to any practical application. There is a particular way of taking some [INAUDIBLE] algebra, and now attaching indices to the [INAUDIBLE]. OK? Now, people do this all the time-- [INAUDIBLE] out of ordinary [INAUDIBLE]. But there's a specific way that you can do it, a specific [? grading, ?] and a specific funny cubic relation that has to be satisfied, between these [INAUDIBLE] structure constants, know as [INAUDIBLE], that allows one of these infinite dimensional objects to be a particularly special, nice one called Yangian.
AUDIENCE: [INAUDIBLE].
NIMA ARKANI-HAMED: I don't even know what the whole, big space is called. But--
AUDIENCE: But the [INAUDIBLE] of this [INAUDIBLE].
NIMA ARKANI-HAMED: Yes, the [INAUDIBLE] of these gigantic, big algebras are these things which [INAUDIBLE] Yangians. In practice, they're the thing that you get by commuting conformal and dual conformal invariances. OK? I could tell you, but I'd have to learn it myself.
And I think-- one thing, you see, amongst other things, we're going to be writing down a formula that generates all the Yangian invariance. And I'm very proud to have written this formula down without actually knowing what the Yangian algebra is. So there's an official definition, but anyway, this is what it is in practice.
OK, so the suspense grows. Right? So the theory [INAUDIBLE] other symmetries. Now, how did people see that the symmetry with there, in tree amplitudes? Well, they took their [INAUDIBLE] the tree amplitude, like the BCFW formula. And they found, really remarkably, that the individual terms in BCFW were individually both conformal and dual conformal [INAUDIBLE]. OK? So that was really, really interesting, right?
Because now, these terms seem to be like building blocks that know about all the symmetries of the theory. This now explains why they weren't local. They couldn't possibly be local, because they're serving two masters. Right? They're trying to make conformal invariance in ordinary space-time and the dual space-time manifest at the same time. Of course they can't be local. OK? But they are showing that the symmetry exists.
So A, they're coming from someplace that knows that it's possible to be invariant under both [? things. ?] But B, they have to be put together in a really interesting way, in order for the non-local poles to cancel, and be left with local poles. So you see, going from all these highfalutin, somewhat vague, [INAUDIBLE] questions about the emergence of space-time, all of a sudden, we're left with a very, very sharp set of questions. OK?
Where did the BCFW terms come from? Is there a generated function [INAUDIBLE]? Is there something that spits them all out? And secondly, how can you understand-- how do you see that when you sum them up, all the non-local terms cancel? That's the rule that tells you where local space-time physics comes from, right-- because the individual ones don't do it. We'll see it in more detail when you actually see what the expressions look like.
But the remarkable thing is that, as I said, the fact that these non-local poles cancel, turn into identities-- identities between rational functions-- you know, functions of ratios of [INAUDIBLE], OK-- ratios of invariance built [INAUDIBLE]. These rational function identities look insane. You would never think that they're true. There are really remarkable-- many, many [? term ?] identities that have to be satisfied. And yet, they are satisfied. OK?
So now it's possible, as we'll discuss, it's possible to drive BCFW from quantum-field theory, and that [INAUDIBLE] of these facts, because we know these non-local poles aren't there. But it's a really roundabout way of just generating these very sharp identities. So the question arises, is there some more direct way of understanding where these identities come from? And whatever that way is has got to give us some insight into where local space-time physics comes from, OK-- and a formulation that gives us these [INAUDIBLE].
All right. So I'm going to greatly accelerate the story of the history. So this is substantially what many of the introductions of the various talks that I, and many other people, have given over the last number of years look like-- even a year ago or a year and a half ago.
To all [INAUDIBLE] theory, to all [INAUDIBLE], which gives you, not the scattering amplitude directly, but actually the thing that you integrate to get the scattering amplitude-- the [INAUDIBLE] of the scattering amplitude. As we'll discuss, it's a very important feature of planar theories that the [INAUDIBLE] scattering amplitude is a well-defined notion. It's not a well-defined notion for a general. But for planar theory, it is a well-defined notion.
So there's a machine that spits out the [INAUDIBLE] of scattering amplitudes. OK? It does it in a way that makes the Yangian invariance completely manifest. Each term's manifestly Yangian invariant. And it actually generalizes the BCFW recursion relations for tree amplitudes to [INAUDIBLE].
So the fact that there is some-- it doesn't say the word [INAUDIBLE] once. There isn't the word [INAUDIBLE] invariance once. OK? And furthermore, of course, it's really, really fast. So you can use it to compute not just tree amplitudes, but one-loop, two-loop, three-loop amplitudes. It's just a matter of hitting [INAUDIBLE] and waiting a few microseconds, and you have the answer.
But more important than that is the fact that it's really showing that this picture exists. There is a dual theory. The Yangian invariance is manifest. Individual terms aren't local. And the 10% which is missing is a completely independent, completely autonomous understanding of why you selected these particular objects to put together, to make local physics, although even that's understood, and is being more and more understood now, for a variety of-- well, that's the one thing which is left.
The thing that ends up making-- ah, and another-- so that's some work that just came out, around a month ago now-- a month, month and a half ago. And in the intervening weeks, various other things have been completely cleaned up and understood now, finally. For example, there was that picture of the Wilson loop, that I told you about. And for a long time, people wanted to know precisely what the object was-- the pole supersymmetric object. What's the super generalization of this Wilson loop that allows you to put in [INAUDIBLE] information as well? OK?
That object has been discovered. So what [INAUDIBLE] is-- what its name is, in ordinary space-time-- it's scattering amplitude. It's named in the dual space-time as a certain, specific sort of Wilson loop. The fact that this Wilson loop and the amplitude are the same is understood. So large-- and all of these equalities are understood and established at the level of the [INAUDIBLE]. Right?
So that's a significant development, because we now have in our hands gigantic amounts of data. [INAUDIBLE] know the whole answer. And remarkably, there is evidence that not just the [INAUDIBLE] is simple, but the integrals are simple. That's yet another story. And many of these things end up being related to an observation made a little over a year ago, that everything about [INAUDIBLE] scattering amplitudes seems to be associated with a certain [INAUDIBLE], over the space of k-dimensional planes in n dimensions. OK?
So it's an object that mathematicians have loved and talked about for 150 years, the space of k-dimensional planes in n dimensions, known as the [INAUDIBLE]. And mathematicians have loved playing around in this space for all this time. It's one of the classic areas. You open up any undergraduate textbook in [INAUDIBLE] geometry, and chapter 4, there's some discussion of the [INAUDIBLE]. You talk about projected the spaces. You would talk about [INAUDIBLE].
But anyway, it's a very specific [INAUDIBLE] over the [INAUDIBLE]. But what these words have to do with scattering amplitudes is not obvious. And I'll make it obvious tomorrow, but this was one of the remarkable things-- is that such a very basic, and deep, and bold, and beautiful part of math, that had never had anything to do with physics in any particularly important way, was actually sitting there, smack-heart-dab in the middle of the structure of gauge theory scattering amplitudes. And you'll see why it is that this [INAUDIBLE] formulation completely liberates you from any particular space-time description. OK?
So the object you're talking about is some [INAUDIBLE]. So we ask the question, what would emergent space-time look like? This is not the question that you can ask abstractly. You need to have an example. You need to look at what a possible answer looks like.
And this is a possible answer. I don't if it's the whole answer. I don't know if it's the relevant answer. But it's a possible answer-- that there is some abstract object, which is a [INAUDIBLE] for this totally auxiliary space. And interesting topological properties of the contour encode space-time locality and unitarity. Now that we understand all loops, we can understand that both the unitarity and locality are encoded in topological properties of the allowed contours of [INAUDIBLE].
This [INAUDIBLE] picture ends up being very closely related, in a certain way of thinking about it, to the twistor string theory. OK? So it contains the twistor string theory. It ends up-- it contains the BCFW recursion relations. That's one of the things that inspired it. But it also contains the CSW rules. It contains-- in different representations of this contour, it contains-- essentially, every way anyone had ever seen to write down scattering amplitudes are seen as different contour choices, which can be smoothly deformed into each other on one basic, algebraic, geometric object. And that's what I will spend some time explaining, over the course of the rest of the lectures.
It's also associated with a new picture for what loops are, which, as you'll see, suggests that there's really one object, depending on perhaps an infinite number of particles. And there's canonical rules for removing particles. It's as if you're not looking at them. You're hiding them. You don't want to see what all these other particles are doing. There's some simple canonical rules for removing particles.
And if you do them one at a time-- if you remove particles one at a time-- nothing interesting happens. You would go from a big tree amplitude down to a little tree amplitude. But it turns out that if you remove one particle, then the next, there is essentially a unique second way of removing pairs of particles, that entangles them in a specific way-- not standard quantum entanglement-- entangles it in a specific way I'll tell you about. You're sort of integrating particle one and two, and there is an entangled [INAUDIBLE] that you can choose to remove both of them.
That operation is where loop amplitudes come from. All right? So in a very specific sense, hiding pairs of particles is where quantum mechanics is coming from. OK? So as I said, there are still a number of things left to much more deeply understand what this object is. But all the data is in front of us, and I think that there will be lots and lots of progress in the next little while.
So that's why-- that's the plan for the remainder of the lecture. I want to tell you what the story is, essentially, of the [INAUDIBLE] super [INAUDIBLE], making all the symmetries manifest, making this connection to this underlying [INAUDIBLE] structure manifest, and drawing you some pictures, in the simplest cases-- where we [? understand things ?] perfectly well-- of, ultimately, what this invariant thing is, that the amplitudes are. They're, roughly speaking, like volumes in some abstract space.
And depending on how you-- they're volumes of some polytopes in some abstract space. And depending on how you triangulate these polytopes, you get different representations of the amplitude that make different things obvious. OK? Some of those representations are the CSW rules, and they make space-time into some [INAUDIBLE]. Some of them are the BCFW rules, and make the Yangian symmetry totally manifest.
It turns out, something that we've discovered recently is that there are other ways of chopping them up, that actually give you new expressions for the scattering amplitudes that are completely local, which are not [INAUDIBLE]. And these end up reproducing-- the one idea from the past that had not been absorbed in this business was the very first set of [INAUDIBLE] that were invented by [INAUDIBLE], know as the [INAUDIBLE] recursion relations.
These were the methods-- much more [INAUDIBLE] diagram based-- that people had used in this heroic earlier epoch of doing the hard calculations. They had managed to get up to six particle amplitudes, seven particle amplitudes, using these recursion relations. Well, we now understand that those expressions are actually yet other ways of slicing up these polytopes. And that geometric picture allows us to generalize the expressions of all [INAUDIBLE] particles, for example. OK?
So I'm giving you a rough idea of what it looks like. Roughly speaking, in one representation, it's some [INAUDIBLE] over an abstract mathematical space. In yet another representation of it, it's some kind of volume of a polytope. In both cases, you have to choose representatives-- a representative of the contour or a representative of a triangulation of the polytope. And different representations make different things manifest. In none of the representations is there [INAUDIBLE], or a gauge symmetry, or anything like that.
OK, so after that very lengthy introduction, I think I'll leave it for tomorrow. We'll start way back down on the ground, and I'll introduce the [INAUDIBLE] variables, the supersymmetric variables-- the nice ways of talking about these [INAUDIBLE] scattering amplitudes, and then I'll begin introducing this [INAUDIBLE]. Thanks a lot.
[APPLAUSE]
AUDIENCE: Can I ask you one thing? So it never mentioned crossing symmetry.
NIMA ARKANI-HAMED: Yeah. Crossing symmetry is always--
AUDIENCE: It's always there.
NIMA ARKANI-HAMED: Yes, it's always there. And in fact, if you like, the existence of crossing symmetry is yet another one of these early indications that there's something wrong in [INAUDIBLE] about scattering amplitudes. Because normally, you'd think about scattering amplitudes as in goes to out. You wouldn't suspect there was such a thing as crossing symmetry. Then you discover it in the answer. Like, oh, look at that! I could just call the [INAUDIBLE] energy negative energy, and [INAUDIBLE] incoming, ah. I get the [INAUDIBLE] just by swinging around. Right?
Of course, it has a pretty quick derivation. OK? But still, it's not [INAUDIBLE] to the structure of the problem. Right? It's yet another thing that's telling you, hold on-- that this is something else. Right? That's right. So from the very beginning, we're imagining all [INAUDIBLE] are incoming, and we don't even say anything about how to interpret the negative energy ones. Later, you can say the negative energy ones are [INAUDIBLE] and so on.
AUDIENCE: So what is your starting point, now that you don't have [INAUDIBLE]? For your starting point, just say, these are the fields?
NIMA ARKANI-HAMED: No, we don't even say the word fields [INAUDIBLE]. These are [INAUDIBLE].
[INTERPOSING VOICES]
I should say one more very important thing, related to this. Some of the motivation and philosophy sounds a lot like the [INAUDIBLE] program. OK? And it shares with the [INAUDIBLE] program the desire to compute the S-matrix. However, but there's two very, very big differences. One of them is practical, and one of them is philosophical.
The practical one is that the [INAUDIBLE] matrix people-- even though their heart was in the right place-- turned out to be focusing on the hardest, worst possible theory to apply their techniques.
AUDIENCE: QCD.
NIMA ARKANI-HAMED: Not QCD-- they didn't even know about QCD. The scattering of scalars, which were massive. OK? The scattering of massive scalars-- they were terrified of massless particles. You open up the analytic S-matrix, everywhere they say there's a mass gap, so we can have our nice, analytic ellipse, and blah, blah, blah. They hated massless particles. They were terrified of [INAUDIBLE]. They said, they're not there anyway, except for the photon and the graviton, but who cares about them?
And the scalars are great, because, oh, every now and then they talk about spin. But they say, oh, it's a bit of a pain in the ass. You've got to put spin in, and OK. So you see, from our point of view, the theory they should have been working on is [INAUDIBLE] or [INAUDIBLE] equals [INAUDIBLE]. But they would never have known that. They would have never known that. That's one.
But two-- another very, very important difference is, their whole attitude was that all you had to do was impose unitarity and causality, and you'd get the right answer. And there would be a unique answer. Right? And this was just sort of crazy from the get go, because there are other theories. Like [? QED ?] is a great theory. You know? There are tons of [INAUDIBLE] you could write down.
They clearly were going to satisfy all of these rules. There was no unique answer. Right? So you don't go in-- and plus, related to that, no one actually knew exactly what the imprint of locality was-- what the imprint of causality was, how to decide these things exactly. In fact, if you look at the old books, what they do is get on their high horse about they're going to compute things with these big principles.
Then [INAUDIBLE] look at the back of the book, at what [INAUDIBLE] diagrams they gave you. Right? Say, oh, these are the kind of singularities that are allowed, because we get them from [INAUDIBLE]. OK, great. Forget that we cheated. Now [INAUDIBLE] we'll apply the same principles. You see? It didn't make any sense.
Our attitude is completely different. It's not that we're going to impose unitarity, and causality, and locality. It's totally the other way around. We're going to find some natural theory that doesn't have space-time in it-- has some other ingredients-- find it, and see them as an output. Exactly-- see causality, and locality, and unitarity as an output, not an input.
AUDIENCE: [INAUDIBLE].
NIMA ARKANI-HAMED: So the philosophies [INAUDIBLE].
AUDIENCE: So what is [INAUDIBLE]?
NIMA ARKANI-HAMED: We will see. We'll see, starting tomorrow. What we'll start doing is just getting a good idea of what these functions are, that we're after-- the simplest way to talk about them. So that's why we'll introduce these [INAUDIBLE] variables and these super variables, that allow us to talk about the super amplitudes nicely.
So finally, there is the collection of functions out there. There are some nice collection of functions that come out of computing [INAUDIBLE] diagrams. You might imagine, could they have come out of somewhere else? OK?
AUDIENCE: [INAUDIBLE] like space-time, or particle--
NIMA ARKANI-HAMED: No, particles are an input. A particle is just the external states are definitely inputs.
AUDIENCE: And space-time would be [INAUDIBLE].
NIMA ARKANI-HAMED: Space-time will be [INAUDIBLE], that's right. And then you'll see-- this is why this is such a fun business. There are no rules. It's just like the [INAUDIBLE] model of physics, except there's data.
[LAUGHTER]
[INAUDIBLE] the world. You look at the structure of gauge scattering amplitudes, and then you force yourself to see-- where could it possibly come from, that's not a [INAUDIBLE]? And you'll see that we'll be directly led to this [INAUDIBLE] picture by focusing on the apparently most boring part of the standard scattering amplitude [? answer-- ?] namely, the momentum-conserving delta function.
We'll stare at the momentum-conserving delta function for quite a while, and we'll realize that it's telling us something. And actually, just reinterpreting the momentum-conserving delta function in a new way, in a couple of steps will lead to this [INAUDIBLE] formula, which-- this one formula that seems to know everything about scattering amplitudes [INAUDIBLE].
And it sounds surprising that you should be able to go so far [? from ?] momentum-conserving delta function, except that the usual thing about dualities-- the things that are trivial from one point of view will be non-trivial from the other point of view, and vice versa. The momentum-conserving delta function is trivial, because it's the one part of standard scattering amplitudes that really damn well knows about the space-time.
Where does it come from? It comes from [INAUDIBLE]. It knows about that x inside. It's [INAUDIBLE]. OK? It's integrating over that whole [INAUDIBLE]. It knows about translations beautifully. OK? And that's where the delta function comes from.
You're not putting it inside of the space-time. That momentum-conserving delta function is [INAUDIBLE]. And so thinking about where it might come from is actually almost the entire story, as we'll see. But as I said, there aren't any rules. So we could have some success with [INAUDIBLE]. At some part of the problem, it's just solved for [INAUDIBLE]. In the planar theory for the [INAUDIBLE], but it's now just really solved [INAUDIBLE].
But what do you do for non-planar theories? What do you do for pure QCD? What do you do for gravity? There's a whole program here to reformulate all of physics, eviscerating the space-time.
And by the way, given that we've done it for this theory, that seems to have this gigantic, [INAUDIBLE] structure-- this huge, infinite dimensional symmetry-- you could really be a pessimist and say, oh, this is so special to [INAUDIBLE]. It's never going to happen for anything else. It's just this accident that you have this infinite dimensional symmetry. That's all it is. Good luck doing [INAUDIBLE] anything else. That's one possible attitude.
Another attitude is, no, maybe there's some hint. Even if it's not exactly there-- if it's broken in some way-- there is this underlying structure controlling every theory, because there has to be, because of these highfalutin reasons about being able to take the correspondence [? limit ?] [? of the ?] theory with gravity, which doesn't have space-time in it, and landing on a theory which shouldn't have space-time in it either. So it's really, I think, right now at a very interesting-- there's a very interesting break point here, because there is an existence proof, basically.
I can hand you a theory that-- up to this 10%, 20%, whatever that's left-- basically fulfills the fantasy. But we need another example. We need to see, can it be done for pure QCD? Can it be done for anything else? I think if you can see that it's also happening with theories that aren't this perfect, crystalline jewel like [INAUDIBLE], then there's really [INAUDIBLE] to do but to just carry it out for everything else-- also, anything for gravity, even for [INAUDIBLE].
I should say that I very, very strongly suspect-- oh, we've had wonderful fun with the [INAUDIBLE]. I really strongly suspect it's a drop in the bucket, compared to what's going on in gravity. And it's just because the [INAUDIBLE] things that have been accomplished for [INAUDIBLE], for [INAUDIBLE] theories-- very basic [INAUDIBLE] things like color stripping-- things that just allow you to talk about the amplitudes in the simplest possible way-- these things have not been done for gravity yet.
So there's still-- it's still at a primitive stage. Even at the level of data, it's at a primitive stage. And some original, initial cleanup work has got to be done. But I'm positive that there's a much richer, and more intricate, and deeper structure for gravity than for [INAUDIBLE].
AUDIENCE: I want to talk about [INAUDIBLE].
NIMA ARKANI-HAMED: Yes.
AUDIENCE: And you talked about the least-action principle, and [INAUDIBLE] principle is it gave you most of the way there, to [INAUDIBLE]. But there still [INAUDIBLE] in this small, additional [INAUDIBLE]. So in terms of what you've done, [INAUDIBLE] found a formulation that's equivalent to, say, [INAUDIBLE]. Are there any hints yet about what an additional [INAUDIBLE] set might be?
NIMA ARKANI-HAMED: I honestly think we don't understand it well enough yet to know. You see, it's like-- I think our understanding is like-- and I think I really want to stress this point. I think we've established a kind of [INAUDIBLE]. But even for [INAUDIBLE], [INAUDIBLE], everything, just the-- we have not extracted the lesson yet. OK?
We know how to do the calculations. We can see the symmetry. We're a long way-- we've severed the ropes that were connecting us to quantum field theory. OK? We're definitely doing things without quantum field theory now-- standard quantum field theory. But I don't think we, yet, really understand. We understand very deeply the object, so that there is-- this last 10% or 20% is going to be very, very important for actually-- and I think until we do, I can't really give a good answer.
There are some vague hints. There are some very, very, vague hints. For example, these [INAUDIBLE] naturally live inside bigger spaces. And so you might imagine-- but this is the kind [INAUDIBLE] often useless mathematical speculation that goes nowhere. But you might imagine that there's some generalization of what we're doing, that will allow fluctuations off the [INAUDIBLE] subsurface, into this bigger space. That might have something to do with gravity.
But this is to obstruct. You'll have to see what it actually looks like. And I don't think I said it, but just [INAUDIBLE], it turns out that using twistor space is just the best way of talking about scattering amplitudes. And in this dual space, there's an analog of the twistor space, known as momentum twistor space, which is the best way of talking about scattering amplitudes.
So twister space is kinematics here. You know? It's just the basis. It's the basis to use is discussing the external particles. It just turns out to be a shockingly powerful and useful basis, that allows you to see symmetry that you would not otherwise see. And I should have also mentioned-- [INAUDIBLE] will come up again and again-- but there are many-- the heroes in this business, early on-- of course, there's the [INAUDIBLE] people.
There was Lance [INAUDIBLE], [? David ?] [INAUDIBLE], Edward, obviously. Freddy [INAUDIBLE] name is all over the story, post-2003. I don't think there's one development that [INAUDIBLE]. A lot of young people-- names like James [INAUDIBLE], [? Johannes ?] [INAUDIBLE], Dave [INAUDIBLE]-- Oxford people, [? Lionel ?] [? Mason ?]. And my favorite of this whole crowd of people is a student of Penrose's named Andrew Hodges, who just turned 60 this year, and who has been working on the subject his entire career.
So Penrose, in 1970-- well, even earlier than that, but very clearly in 1972, saw a connection between his twistors and [INAUDIBLE] diagrams. And he invented something called twistor diagrams, which looked to him a very natural analog of [INAUDIBLE] diagrams. And they were complicated [INAUDIBLE] in twistor space. And he went on and on, saying that these things were very important.
Certainly, no particle physicist payed any attention to him whatsoever. Andrew Hodges worked on this problem, from back then. He was, I think one of Penrose's first graduate students. And he worked on it, and worked on it, and worked on it. And he made huge amounts of progress, understanding how to do these [INAUDIBLE] in more and more realistic settings. Again, no one paid the slightest whit of attention to it whatsoever.
His papers sat on my desk for two years, because they have the following bizarre feature-- that I read the introduction. I read the conclusions, and it seemed completely sane, and sober, and totally sensible, and the words were very exciting. And then I read the middle of the paper-- I had not one idea what he was talking about-- zero.
He had these insane-looking pictures. It wasn't even done in [INAUDIBLE]. It was done in Word. And everything, other than the intro and the conclusions, and the fact that he was a doctor, and Penrose's student-- all these things would say, crackpot. OK? But check the intro and conclusions. They were totally sensible. Right?
So as it turned out, he was dead right about everything-- just every single thing. And realizing this is kind of like-- I still have a lot of emotion about it, because it's like a [? standard ?] conversion story. Right? You think, oh, crackpot, crackpot, crackpot, and then it works for you once. In the midst of-- we rediscovered many of these things. But it was painful going for a while, until we saw that-- we started realizing, [INAUDIBLE].
And what he realized-- actually, I didn't mention one thing, interesting sociologically. But [? Witten's ?] paper came out. People got extremely excited about twistors for six months. And then it became clear that the twistor string theory-- it became clear, it still hasn't been proven, by the way-- but people thought that it wouldn't work at loop level, and it died. This is a classic string theory fad, OK-- not amongst great string theorists, but amongst the unwashed masses. OK?
So it died. It died just because of that reason. And I think there was a big sigh of relief that no one had to learn all this weird twistor stuff. OK? And then what came out of it were all these computational tools for [INAUDIBLE], CSW. And then people gave field theory derivations of them, so it will all, oh, [? Witten's ?] work inspired all these things. But really, secretly, it's all quantum field theory. It's all fantastic and great, but forget about all this twistor stuff.
Right at the moment twistors died, there was a conference. I wasn't in the field then, but basically, the moment had died. Andrew Hodges gave a talk at [INAUDIBLE] where he said, no, no, no. It's totally the other way around. These twistors are the natural home for the BCFW recursion relations. They're actually formulated in twistor space.
That seemed completely nuts, because this is really-- as I said, at the time, people were breathing this big sigh of relief they didn't have to learn twistors. And in fact, BCFW had a beautiful, classic field theory derivation that's in [INAUDIBLE] textbook now. It's textbook material. It's just beautiful. It comes from quantum field theory-- just great, just absolutely beautiful.
Not one reason to suspect it has anything to do with twistors, except Hodges, who said, this thing I've been doing for 30 years-- these twistor diagrams-- they aren't [INAUDIBLE] diagrams. They were BCFW terms the whole time. OK? So they were really computing BCFW terms. And I got it now. This is a great. Guys, this is amazing. Right?
Nobody paid any attention. OK? They were too busy breathing a big sigh of relief. Then we realized, in a very circuitous way, that BCFW was naturally formulated in twistor space. So this was, at least for us, the sort of rebirth of twistor ideas.
And then we started actually trying to do calculations. And the calculations got long and messy, so we started inventing some diagrams to keep track of things. And the diagrams were messy. And at some point we realized, wait a minute. These diagrams look like Hodge's diagrams.
So we went to his papers. And sure enough, they started looking similar, except he was doing things much more intelligently than we were. So the conversion experience was just saying, assume he's right. Let's see. We want to do this [INAUDIBLE] particle amplitude. What would we do to make it look like his picture? [INAUDIBLE] no idea, the words he's saying-- nothing. But we said, just kind of go to that target.
All of a sudden, awesome things started happening-- the sixth particle, the seventh particle. And it's completely clear he knew what he was doing. Actually, by this point, there was no rational reason for people to not believe him, because while early on, he wasn't getting any results that other people didn't know. So you could always say, yeah, yeah. He's just BS-ing, drawing all these diagrams, then borrowing someone else's answer. By the last paper that he wrote in this period, he was just doing computations no one had ever done before, getting the right answers, and so on.
So there was something going on. But anyway, he shortly after invented this idea of momentum twistor space, to understand dual conformal symmetry. And all of these developments in the past year and a half have been going, really-- there's a small group of people-- Oxford people, [INAUDIBLE] people, a lot of Europeans-- but it's been a big collective effort, where everyone has seen one or other big particle [INAUDIBLE]. But it's wonderful that these Oxford guys were not smoking crack after all, even though it took a long, long, long time to [INAUDIBLE]. OK.
[APPLAUSE]

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The first in a 5-part series of technical lectures on scattering amplitudes given by Prof. Arkani-Hamed in parallel to his Messenger lectures on fundamental physics at Cornell University. The focus is application to N=4 supersymmetric Yang-Mills Theory.